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      <doc-data|<doc-title|Chapter 3: Differentiation Rules>>

      <\table-of-contents|toc>
        <vspace*|1fn><with|font-series|bold|math-font-series|bold|1<space|2spc>Derivatives
        of Polynomials and Exponential Functions>
        <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
        <no-break><pageref|auto-1><vspace|0.5fn>

        <with|par-left|1tab|1.1<space|2spc>Power functions
        <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
        <no-break><pageref|auto-2>>

        <with|par-left|1tab|1.2<space|2spc>Linear combination
        <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
        <no-break><pageref|auto-3>>

        <with|par-left|1tab|1.3<space|2spc>Exponential functions
        <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
        <no-break><pageref|auto-4>>

        <vspace*|1fn><with|font-series|bold|math-font-series|bold|2<space|2spc>The
        Product and Quotient Rules> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
        <no-break><pageref|auto-5><vspace|0.5fn>

        <with|par-left|1tab|2.1<space|2spc>The product rule
        <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
        <no-break><pageref|auto-6>>

        <with|par-left|1tab|2.2<space|2spc>The quotient rule
        <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
        <no-break><pageref|auto-7>>

        <vspace*|1fn><with|font-series|bold|math-font-series|bold|3<space|2spc>Derivatives
        of Trigonometric Functions> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
        <no-break><pageref|auto-8><vspace|0.5fn>

        <with|par-left|1tab|Derivative of
        <with|color|black|font-family|rm|<with|math-display|true|<with|mode|math|<with|font-base-size|12|sin
        x>>>> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
        <no-break><pageref|auto-9>>

        <vspace*|1fn><with|font-series|bold|math-font-series|bold|4<space|2spc>The
        Chain Rule> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
        <no-break><pageref|auto-10><vspace|0.5fn>

        <vspace*|1fn><with|font-series|bold|math-font-series|bold|5<space|2spc>Implicit
        differentiation> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
        <no-break><pageref|auto-11><vspace|0.5fn>

        <vspace*|1fn><with|font-series|bold|math-font-series|bold|6<space|2spc>Derivatives
        of Logarithmic Functions> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
        <no-break><pageref|auto-12><vspace|0.5fn>

        <with|par-left|1tab|6.1<space|2spc>Logarithmic differentiation
        <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
        <no-break><pageref|auto-13>>

        <vspace*|1fn><with|font-series|bold|math-font-series|bold|7<space|2spc>Rates
        of Change in the Economics and Social Sciences>
        <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
        <no-break><pageref|auto-14><vspace|0.5fn>

        <vspace*|1fn><with|font-series|bold|math-font-series|bold|8<space|2spc>skip>
        <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
        <no-break><pageref|auto-15><vspace|0.5fn>

        <vspace*|1fn><with|font-series|bold|math-font-series|bold|9<space|2spc>skip>
        <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
        <no-break><pageref|auto-16><vspace|0.5fn>

        <vspace*|1fn><with|font-series|bold|math-font-series|bold|10<space|2spc>Linear
        Approximations and Differentials>
        <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
        <no-break><pageref|auto-17><vspace|0.5fn>

        <with|par-left|1tab|10.1<space|2spc>Linear approximations
        <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
        <no-break><pageref|auto-18>>

        <with|par-left|1tab|10.2<space|2spc>Differentials
        <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
        <no-break><pageref|auto-19>>
      </table-of-contents>

      <section|Derivatives of Polynomials and Exponential Functions>

      <subsection|Power functions>

      <\itemize>
        <item>If <math|f<around*|(|x|)>=c>, where <math|c> is a constant.
        Then

        <\equation*>
          f<rprime|'><around*|(|x|)>=lim<rsub|h\<rightarrow\>0>
          <dfrac|f<around*|(|x+h|)>-f<around*|(|x|)>|h>=lim<rsub|h\<rightarrow\>0>
          <dfrac|c-c|h>=0.
        </equation*>

        <item>If <math|f<around*|(|x|)>=x>, then

        <\equation*>
          f<rprime|'><around*|(|x|)>=lim<rsub|h\<rightarrow\>0>
          <dfrac|f<around*|(|x+h|)>-f<around*|(|x|)>|h>=lim<rsub|h\<rightarrow\>0>
          <dfrac|<around*|(|x+h|)>-x|h>=1.
        </equation*>

        <item>If <math|f<around*|(|x|)>=x<rsup|2>>, then

        <\equation*>
          f<rprime|'><around*|(|x|)>=lim<rsub|h\<rightarrow\>0>
          <dfrac|f<around*|(|x+h|)>-f<around*|(|x|)>|h>=lim<rsub|h\<rightarrow\>0>
          <dfrac|<around*|(|x+h|)><rsup|2>-x<rsup|2>|h>
        </equation*>

        <\equation*>
          =lim<rsub|h\<rightarrow\>0> <dfrac|2*x*h+h<rsup|2>|h>=lim<rsub|h\<rightarrow\>0>
          <around*|(|2x+h|)>=2*x.
        </equation*>

        <item>In general, if <math|f<around*|(|x|)>=x<rsup|n>,n\<in\>\<bbb-N\>>,
        then

        <\equation*>
          f<rprime|'><around*|(|x|)>=n*x<rsup|n-1>.
        </equation*>

        Hint: use the binomial expansion

        <\equation*>
          <around*|(|a+b|)><rsup|n>=a<rsup|n>+n*a<rsup|n-1>*b+<dfrac|n<around*|(|n-1|)>|2>*a<rsup|n-2>*b<rsup|2>+\<cdots\>+b<rsup|n>=<big|sum><rsub|k=0><rsup|n><matrix|<tformat|<table|<row|<cell|n>>|<row|<cell|k>>>>>*a<rsup|k>*b<rsup|n-k>,
        </equation*>

        where

        <\equation*>
          <matrix|<tformat|<table|<row|<cell|n>>|<row|<cell|k>>>>>=<dfrac|n!|k!*<around*|(|n-k|)>!>.
        </equation*>

        <item>The above results can be generalized to
        <math|f<around*|(|x|)>=x<rsup|a>,a\<in\>\<bbb-R\>>.\ 

        <\equation*>
          <marked|f<rprime|'><around*|(|x|)>=a*x<rsup|a-1>>.
        </equation*>
      </itemize>

      \;

      From this result we can find the derivative of any polynomial.

      <subsection|Linear combination>

      <\equation*>
        <marked|<around*|[|c<rsub|1>*f<around*|(|x|)>+c<rsub|2>*g<around*|(|x|)>|]><rprime|'>=c<rsub|1>*f<rprime|'><around*|(|x|)>+c<rsub|2>*g<rprime|'><around*|(|x|)>>.
      </equation*>

      <\example>
        If <math|f<around*|(|x|)>=x<rsup|100>+5*x<rsup|3>+6*x-12,>then

        <\equation*>
          f<rprime|'><around*|(|x|)>=100*x<rsup|99>+15*x<rsup|2>+6
        </equation*>
      </example>

      <subsection|Exponential functions>

      Let <math|f<around*|(|x|)>=b<rsup|x>,b\<gtr\>0>. Then

      <\eqnarray*>
        <tformat|<table|<row|<cell|f<rprime|'><around*|(|x|)>>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
        <dfrac|f<around*|(|x+h|)>-f<around*|(|x|)>|h>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
        <dfrac|b<rsup|x+h>-b<rsup|x>|h>=lim<rsub|h\<rightarrow\>0>
        <dfrac|b<rsup|x>*<around*|(|b<rsup|h>-1|)>|h>>>|<row|<cell|>|<cell|=>|<cell|b<rsup|x>*lim<rsub|h\<rightarrow\>0>
        <dfrac|*b<rsup|h>-1|h>.>>>>
      </eqnarray*>

      In fact,\ 

      <\equation*>
        lim<rsub|h\<rightarrow\>0> <dfrac|*b<rsup|h>-1|h>=f<rprime|'><around*|(|0|)>
      </equation*>

      if it exists, and

      <\equation*>
        f<rprime|'><around*|(|x|)>=f<rprime|'><around*|(|0|)>*b<rsup|x>.
      </equation*>

      In fact,\ 

      <\equation*>
        <marked|<around*|(|b<rsup|x>|)><rprime|'>=<around*|(|ln
        b|)>*b<rsup|x>>.
      </equation*>

      In particular,

      <\equation*>
        <marked|<around*|(|e<rsup|x>|)><rprime|'>=e<rsup|x>>.
      </equation*>

      <\question>
        Prove the limit exists. Hint: tranform the limit to the form
        containing the term <math|lim<rsub|x\<rightarrow\>\<infty\>><around*|(|1+<dfrac|1|x>|)><rsup|x>>.
      </question>

      <hlink|Click Here|https://planetmath.org/convergenceofthesequence11nn>
      to see a proof that the limit of the sequence
      <math|<around*|(|1+<dfrac|1|n>|)><rsup|n>>exists as
      <math|n\<rightarrow\>\<infty\>>.

      \;

      <\definition>
        The <strong|relative rate of change> of a function
        <math|f<around*|(|x|)>> is

        <\equation*>
          <dfrac|f<rprime|'><around*|(|x|)>|f<around*|(|x|)>>,
        </equation*>

        and the <strong|percentage rate of change> is

        <\equation*>
          100*<dfrac|f<rprime|'><around*|(|x|)>|f<around*|(|x|)>>.
        </equation*>
      </definition>

      \;

      <section|The Product and Quotient Rules>

      <subsection|The product rule>

      \;

      Motivation: How to find the derivative function of
      <math|f<around*|(|x|)>=x<rsup|2>*e<rsup|x>>?

      \;

      <\theorem>
        <dueto|Product Rule>If <math|f> and <math|g> are both differentiable
        at <math|x>, then <math|f*g> is differentiable at <math|x>. Moreover,

        <\equation*>
          <around*|(|f*g|)><rprime|'><around*|(|x|)>=f<rprime|'><around*|(|x|)>*g<around*|(|x|)>+f<around*|(|x|)>*g<rprime|'><around*|(|x|)>.
        </equation*>
      </theorem>

      <\proof>
        From the definition,

        <\eqnarray*>
          <tformat|<table|<row|<cell|<around*|(|f*g|)><rprime|'><around*|(|x|)>>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
          <dfrac|f<around*|(|x+h|)>*g<around*|(|x+h|)>-f<around*|(|x|)>*g<around*|(|x|)>|h>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
          <dfrac|f<around*|(|x+h|)>*g<around*|(|x+h|)><with|color|blue|-f<around*|(|x|)>*g<around*|(|x+h|)>+f<around*|(|x|)>*g<around*|(|x+h|)>>-f<around*|(|x|)>*g<around*|(|x|)>|h>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
          <dfrac|g<around*|(|x+h|)>*<around*|[|f<around*|(|x+h|)>-f<around*|(|x|)>|]>+f<around*|(|x|)>*<around*|[|g<around*|(|x+h|)>-g<around*|(|x|)>|]>|h>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
          <dfrac|g<around*|(|x+h|)>*<around*|[|f<around*|(|x+h|)>-f<around*|(|x|)>|]>|h>+lim<rsub|h\<rightarrow\>0>
          <dfrac|f<around*|(|x|)>*<around*|[|g<around*|(|x+h|)>-g<around*|(|x|)>|]>|h>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
          g<around*|(|x+h|)>*lim<rsub|h\<rightarrow\>0>
          <dfrac|f<around*|(|x+h|)>-f<around*|(|x|)>|h>+lim<rsub|h\<rightarrow\>0>
          f<around*|(|x|)>*lim<rsub|h\<rightarrow\>0>
          <dfrac|g<around*|(|x+h|)>-g<around*|(|x|)>|h>>>|<row|<cell|>|<cell|=>|<cell|g<around*|(|x|)>*f<rprime|'><around*|(|x|)>+f<around*|(|x|)>*g<rprime|'><around*|(|x|)>.>>>>
        </eqnarray*>

        \;
      </proof>

      Intuitive explanation

      <\wide-block>
        <tformat|<cwith|1|1|2|2|cell-valign|c>|<cwith|1|1|1|1|cell-valign|c>|<table|<row|<\cell>
          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|png>|0.33par|||>
        </cell>|<\cell>
          Let <math|u=f<around*|(|x|)>,v=g<around*|(|x|)>>. For small change
          <math|\<Delta\>x>, let <math|\<Delta\>u,\<Delta\>v> be the change
          in <math|u,v>, respectively. Let <math|\<Delta\><around*|(|u*v|)>>
          be the change in <math|u*v>.

          <\eqnarray*>
            <tformat|<table|<row|<cell|\<Delta\><around*|(|u*v|)>>|<cell|=>|<cell|<around*|(|u+\<Delta\>u|)>*<around*|(|v+\<Delta\>v|)>-u*v>>|<row|<cell|>|<cell|=>|<cell|u*v+u*\<Delta\>v+v*\<Delta\>u+\<Delta\>u*\<Delta\>v-u*v>>|<row|<cell|>|<cell|=>|<cell|u*\<Delta\>v+v*\<Delta\>u+\<Delta\>u*\<Delta\>v>>>>
          </eqnarray*>

          <\eqnarray*>
            <tformat|<table|<row|<cell|<dfrac|\<Delta\><around*|(|u*v|)>|\<Delta\>x>>|<cell|=>|<cell|<dfrac|u*\<Delta\>v+v*\<Delta\>u+\<Delta\>u*\<Delta\>v|\<Delta\>x>>>|<row|<cell|>|<cell|=>|<cell|u*<dfrac|\<Delta\>v|\<Delta\>x>+v*<dfrac|\<Delta\>u|\<Delta\>x>+\<Delta\>u*<dfrac|\<Delta\>v|\<Delta\>x>>>|<row|<cell|>|<cell|\<rightarrow\>>|<cell|u*v<rprime|'>+v*u<rprime|'>+0<space|1em>as<space|1em>\<Delta\>x\<rightarrow\>0>>|<row|<cell|>|<cell|=>|<cell|u*v<rprime|'>+v*u<rprime|'>>>>>
          </eqnarray*>
        </cell>>>>
      </wide-block>

      \;

      <\example>
        \;

        <\enumerate-alpha>
          <item>If <math|f<around|(|x|)>=x*e<rsup|x>>, find
          <math|f<rprime|'><around|(|x|)>>.\ 

          <\equation*>
            <around*|(|x*e<rsup|x>|)><rprime|'>=x<rprime|'>*e<rsup|x>+x*<around*|(|e<rsup|x>|)><rprime|'>=e<rsup|x>+x*e<rsup|x>=e<rsup|x>*<around*|(|1+x|)>.
          </equation*>

          <item>Find the <math|n>th derivative,
          <math|f<rsup|<around|(|n|)>><around|(|x|)>>.

          <\eqnarray*>
            <tformat|<table|<row|<cell|<around*|(|x*e<rsup|x>|)><rprime|''>>|<cell|=>|<cell|<around*|[|e<rsup|x>*<around*|(|1+x|)>|]><rprime|'>>>|<row|<cell|>|<cell|=>|<cell|e<rsup|x><around*|(|1+x|)>+e<rsup|x>=e<rsup|x>*<around*|(|2+x|)>.>>>>
          </eqnarray*>

          Repeat this process one more time, we get\ 

          <\equation*>
            <around*|(|x*e<rsup|x>|)><rprime|'''>=e<rsup|x>*<around*|(|3+x|)>.
          </equation*>

          In general

          <\equation*>
            f<rsup|<around*|(|n|)>>=e<rsup|x>*<around*|(|n+x|)>.
          </equation*>

          <\exercise>
            Prove the general result using mathematical induction.
          </exercise>
        </enumerate-alpha>
      </example>

      <subsection|The quotient rule>

      <\theorem>
        <dueto|Quotient Rule>If <math|f> and <math|g> are both differentiable
        at <math|x> and <math|g<around*|(|x|)>\<neq\>0>, then <math|f*g> is
        differentiable at <math|x>. Moreover,

        <\equation*>
          <around*|(|<dfrac|f|g>|)><rprime|'><around*|(|x|)>=<dfrac|f<rprime|'>*<around*|(|x|)>g<around*|(|x|)>-f*<around*|(|x|)>g<rprime|'><around*|(|x|)>|g<rsup|2><around*|(|x|)>>.
        </equation*>
      </theorem>

      \;

      <\exercise>
        Prove the quotient rule using the definition of derivative.
      </exercise>

      <\example>
        Let <math|y=<frac|x<rsup|2>+x-2|x<rsup|3>+6>>. Then

        <\eqnarray*>
          <tformat|<table|<row|<cell|y<rprime|'>>|<cell|=>|<cell|<frac|<around*|(|x<rsup|2>+x-2|)><rprime|'>*<around*|(|x<rsup|3>+6|)>-<around*|(|x<rsup|2>+x-2|)>*<around*|(|x<rsup|3>+6|)><rprime|'>|<around*|(|x<rsup|3>+6|)><rsup|2>>>>|<row|<cell|>|<cell|=>|<cell|<frac|<around*|(|2*x+1|)>*<around*|(|x<rsup|3>+6|)>-<around*|(|x<rsup|2>+x-2|)>*<around*|(|3*x<rsup|2>|)>|<around*|(|x<rsup|3>+6|)><rsup|2>>>>|<row|<cell|>|<cell|=>|<cell|<frac|2*x<rsup|4>+x<rsup|3>+12*x+6-<around*|(|3*x<rsup|4>+3*x<rsup|3>-6*x<rsup|2>|)>|<around*|(|x<rsup|3>+6|)><rsup|2>>>>|<row|<cell|>|<cell|=>|<cell|<frac|-x<rsup|4>-2*x<rsup|3>+6*x<rsup|2>+12*x+6|<around*|(|x<rsup|3>+6|)><rsup|2>>.>>>>
        </eqnarray*>
      </example>

      \;

      <section|Derivatives of Trigonometric Functions>

      <subsection*|Derivative of <math|sin x>>

      By the definition

      <\eqnarray*>
        <tformat|<table|<row|<cell|<around*|(|sin
        x|)><rprime|'>>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
        <dfrac|sin<around*|(|x+h|)>-sin<around*|(|x|)>|h>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
        <dfrac|sin<around*|(|x|)>*cos<around*|(|h|)>+cos<around*|(|x|)>*sin<around*|(|h|)>-sin<around*|(|x|)>|h>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
        <dfrac|sin<around*|(|x|)>*<around*|[|cos<around*|(|h|)>-1|]>+cos<around*|(|x|)>*sin<around*|(|h|)>|h>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
        <dfrac|sin<around*|(|x|)>*<around*|[|cos<around*|(|h|)>-1|]>|h>+lim<rsub|h\<rightarrow\>0>
        <dfrac|cos<around*|(|x|)>*sin<around*|(|h|)>|h>>>|<row|<cell|>|<cell|=>|<cell|sin<around*|(|x|)>*lim<rsub|h\<rightarrow\>0>
        <dfrac|cos<around*|(|h|)>-1|h>+cos<around*|(|x|)>*lim<rsub|h\<rightarrow\>0>
        <dfrac|sin<around*|(|h|)>|h>>>|<row|<cell|>|<cell|=>|<cell|cos<around*|(|x|)>.>>>>
      </eqnarray*>

      <\wide-block>
        <tformat|<cwith|1|1|2|2|cell-valign|c>|<cwith|1|1|1|1|cell-valign|c>|<table|<row|<\cell>
          <with|gr-mode|<tuple|edit|line>|gr-frame|<tuple|scale|1.00003cm|<tuple|0.5gw|0.5gh>>|gr-geometry|<tuple|geometry|0.4par|0.6par|center>|gr-grid|<tuple|cartesian|<point|0|0>|1>|gr-grid-old|<tuple|cartesian|<point|0|0>|1>|gr-edit-grid-aspect|<tuple|<tuple|axes|none>|<tuple|1|none>|<tuple|4|none>>|gr-edit-grid|<tuple|cartesian|<point|0|0>|1>|gr-edit-grid-old|<tuple|cartesian|<point|0|0>|1>|gr-grid-aspect|<tuple|<tuple|axes|#808080>|<tuple|1|#c0c0c0>|<tuple|4|#e0e0ff>>|gr-grid-aspect-props|<tuple|<tuple|axes|#808080>|<tuple|1|#c0c0c0>|<tuple|4|#e0e0ff>>|gr-snap|<tuple|curve
          point|curve-curve intersection>|gr-auto-crop|true|gr-snap-distance|5px|<graphics||<carc|<point|-1.25|1.25>|<point|1.25|-1.25>|<point|-1.25|-1.25>>|<line|<point|0|0>|<point|1.76776695296637|0.0>>|<math-at|A|<point|1.85482913922745|-0.111883894438878>>|<math-at|B|<point|0.8777543511513498|0.9301335317706395>>|<math-at|h|<point|0.5452616202957267|0.07332534071961085>>|<math-at|D|<point|1.7662881710946974|1.3363649726233786>>|<line|<point|1.45083|1.01>|<point|1.7677660586197912|-0.001778199046468032>>|<line|<point|0|0.00356307>|<point|1.7577628339396612|1.225535589607908>>|<line|<point|1.7577628339396612|1.225535589607908>|<point|1.7748135082497336|-0.010638297872340425>>|<math-at|O|<point|-0.4010941311028125|-0.29750995604183605>>>>
        </cell>|<\cell>
          First, assume <math|0\<less\>h\<less\><dfrac|\<pi\>|2>>

          <\equation*>
            <around*|\||\<Delta\>O*A*B|\|>\<less\><around*|\||<wide|O*A*B|\<invbreve\>>|\|>\<less\><around*|\||\<Delta\>O*A*D|\|>
          </equation*>

          <\equation*>
            <dfrac|1|2>*sin<around*|(|h|)>\<less\><dfrac|1|2>*h\<less\><dfrac|1|2>*tan<around*|(|h|)>
          </equation*>

          <\equation*>
            \<Rightarrow\><space|1em>*sin<around*|(|h|)>\<less\>h\<less\>tan<around*|(|h|)>
          </equation*>

          <\equation*>
            \<Rightarrow\><space|1em>1\<less\><dfrac|h|sin
            h>\<less\><dfrac|1|cos h>
          </equation*>

          <\equation*>
            \<Rightarrow\><space|1em>cos h\<less\><dfrac|sin h|h>\<less\>1
          </equation*>

          <\equation*>
            \<Rightarrow\><space|1em>lim<rsub|h\<rightarrow\>0> cos
            h\<leqslant\>lim<rsub|h\<rightarrow\>0> <dfrac|sin
            h|h>\<leqslant\>lim<rsub|h\<rightarrow\>0> 1
          </equation*>

          <\equation*>
            \<Rightarrow\><space|1em>1\<leqslant\>lim<rsub|h\<rightarrow\>0>
            <dfrac|sin h|h>\<leqslant\>1.
          </equation*>

          By the squeeze theorem, we have

          <\equation*>
            lim<rsub|h\<rightarrow\>0> <dfrac|sin h|h>=1.
          </equation*>

          <\equation*>
            \;
          </equation*>
        </cell>>>>
      </wide-block>

      <\eqnarray*>
        <tformat|<table|<row|<cell|lim<rsub|h\<rightarrow\>0>
        <dfrac|cos<around*|(|h|)>-1|h>>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
        <dfrac|cos<around*|(|h|)>-1|h>*<dfrac|cos<around*|(|h|)>+1|cos<around*|(|h|)>+1>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
        <dfrac|cos<rsup|2><around*|(|h|)>-1|h<around*|(|cos
        h+1|)>>*>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
        <dfrac|-sin<rsup|2> h|h<around*|(|cos
        h+1|)>>*>>|<row|<cell|>|<cell|=>|<cell|-lim<rsub|h\<rightarrow\>0>
        <dfrac|sin<rsup|2> h|h>**lim<rsub|h\<rightarrow\>0> <dfrac|1|cos
        h+1>>>|<row|<cell|>|<cell|=>|<cell|-<dfrac|1|2>*lim<rsub|h\<rightarrow\>0>
        <dfrac|sin<rsup|2> h|h>=-<dfrac|1|2>*lim<rsub|h\<rightarrow\>0>
        <dfrac|sin h|h>*lim<rsub|h\<rightarrow\>0> sin h=0.>>>>
      </eqnarray*>

      For your record

      <\equation*>
        <marked|lim<rsub|x\<rightarrow\>0> <dfrac|sin
        x|x>=1,<space|1em>lim<rsub|x\<rightarrow\>0> <dfrac|1-cos x|x>=0>.
      </equation*>

      <\theorem>
        <dueto|Derivaties of <math|sin> and <math|cos>>

        <\equation*>
          <around*|(|sin x|)><rprime|'>=cos x,<space|1em><around*|(|cos
          x|)><rprime|'>=-sin x.
        </equation*>
      </theorem>

      \;

      Using the quotient rule, we can find the derivatives of other
      trigonometric functions.

      <\equation*>
        <around*|(|tan x|)><rprime|'>=<around*|(|<dfrac|sin x|cos
        x>|)><rprime|'>=<dfrac|cos<around*|(|x|)>*cos<around*|(|x|)>-sin<around*|(|x|)>*<around*|(|-sin
        x|)>|cos<rsup|2> x>=<dfrac|1|cos<rsup|2> x>=sec<rsup|2> x.
      </equation*>

      <\exercise>
        Find the derivative of <math|cot x,sec x,csc x>.
      </exercise>

      <section|The Chain Rule>

      Motivation: If <math|F<around*|(|x|)>=<sqrt|1+x<rsup|2>>>, then
      <math|F<rprime|'><around*|(|x|)>=?>

      \;

      <\theorem>
        <dueto|The Chain Rule>Suppose <math|g> is differentiable at <math|x>
        and <math|f> is differentiable at <math|g<around*|(|x|)>>. Then
        <math|f\<circ\>g> is differentiable at <math|x>, and

        <\equation*>
          <around*|(|f\<circ\>g|)><rprime|'><around*|(|x|)>=f<rprime|'><around*|(|g<around*|(|x|)>|)>*g<rprime|'><around*|(|x|)>.
        </equation*>

        Another way to say this. Suppose <math|y> is a function of <math|x>,
        which is differentiable at <math|a>, and <math|z> is a function of
        <math|y>, which is differentiable at <math|y<around*|(|a|)>>, then
        <math|z> is a function of <math|x>, which is differentiable at
        <math|a>, and

        <\equation*>
          <dfrac|d<hspace|0pt>z|d<hspace|0pt>x>=<dfrac|d<hspace|>z|d<hspace|0pt>y>*<dfrac|d<hspace|0pt>y|d<hspace|0pt>x>
        </equation*>
      </theorem>

      <\proof>
        By the definition,

        <\eqnarray*>
          <tformat|<table|<row|<cell|<around*|(|f\<circ\>g|)><rprime|'><around*|(|x|)>>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
          <dfrac|f<around*|(|g<around*|(|x+h|)>|)>-f<around*|(|g<around*|(|x|)>|)>|h>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
          <dfrac|f<around*|(|g<around*|(|x+h|)>|)>-f<around*|(|g<around*|(|x|)>|)>|h>*<dfrac|g<around*|(|x+h|)>-g<around*|(|x|)>|g<around*|(|x+h|)>-g<around*|(|x|)>>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
          <dfrac|f<around*|(|g<around*|(|x+h|)>|)>-f<around*|(|g<around*|(|x|)>|)>|g<around*|(|x+h|)>-g<around*|(|x|)>>**lim<rsub|h\<rightarrow\>0>
          *<dfrac|g<around*|(|x+h|)>-g<around*|(|x|)>|h>>>|<row|<cell|>|<cell|=>|<cell|f<rprime|'><around*|(|g<around*|(|x|)>|)>*g<rprime|'><around*|(|x|)>.>>>>
        </eqnarray*>

        \;
      </proof>

      <\example>
        Find <math|F<rprime|'><around*|(|x|)>> if
        <math|F<around*|(|x|)>=<sqrt|1+x<rsup|2>>>.
      </example>

      <strong|Answer:> <math|F=f\<circ\>g>, where
      <math|f<around*|(|x|)>=<sqrt|x>,g<around*|(|x|)>=1+x<rsup|2>>.

      <\equation*>
        F<rprime|'><around*|(|x|)>=f<rprime|'><around*|(|g<around*|(|x|)>|)>*g<rprime|'><around*|(|x|)>=<dfrac|1|2*<sqrt|g<around*|(|x|)>>>*2*x=<dfrac|1|<sqrt|1+x<rsup|2>>>.
      </equation*>

      \;

      <\example>
        Differentiate <math|y=<around*|(|x<rsup|3>-1|)><rsup|100>>.
      </example>

      <strong|Answer:>\ 

      <\equation*>
        y<rprime|'>=100*<around*|(|x<rsup|3>-1|)><rsup|99>*<around*|(|3*x<rsup|2>|)>=300*x<rsup|2>*<around*|(|x<rsup|3>-1|)><rsup|99>.
      </equation*>

      \;

      <strong|Generalization>: For example,\ 

      <\equation*>
        <around*|(|f\<circ\>g\<circ\>h|)><rprime|'><around*|(|x|)>=f<rprime|'><around*|(|g<around*|(|h<around*|(|x|)>|)>|)>*g<rprime|'><around*|(|h<around*|(|x|)>|)>*h<rprime|'><around*|(|x|)>.
      </equation*>

      <\example>
        Find <math|f<rprime|'><around*|(|x|)>> if
        <math|f<around*|(|x|)>=2<rsup|<sqrt|3+sin x>>>

        <\equation*>
          f<rprime|'><around*|(|x|)>=<around*|(|ln 2|)>*2<rsup|<sqrt|3+sin
          x>>*<dfrac|cos x|2*<sqrt|3+sin x>>*.
        </equation*>
      </example>

      <section|Implicit differentiation>

      Motivation: How to find <math|y<rprime|'><around*|(|x|)>> if
      <math|e<rsup|x+y>=sin y>?

      <\example>
        Find <math|<around*|\<nobracket\>|<dfrac|d<hspace|0pt>y|d<hspace|0pt>x>|\|><rsub|x=3,y=4><rsub|>>
        if <math|x<rsup|2>+y<rsup|2>=25>.
      </example>

      <with|gr-mode|<tuple|edit|point>|gr-frame|<tuple|scale|1cm|<tuple|0.5gw|0.5gh>>|gr-geometry|<tuple|geometry|1par|0.6par>|gr-grid|<tuple|cartesian|<point|0|0>|1>|gr-grid-old|<tuple|cartesian|<point|0|0>|1>|gr-edit-grid-aspect|<tuple|<tuple|axes|none>|<tuple|1|none>|<tuple|10|none>>|gr-edit-grid|<tuple|cartesian|<point|0|0>|1>|gr-edit-grid-old|<tuple|cartesian|<point|0|0>|1>|gr-auto-crop|true|<graphics||<line|<point|-4|0>|<point|4.6|0.0>>|<line|<point|0|-2.4>|<point|0.0|2.8>>|<carc|<point|-2|0>|<point|2.1|1.38777878078145e-17>|<point|2.0|0.7>>|<point|-2|0>|<point|2.1|6.93889e-18>|<point|1.40949|1.6>|<point|1.2|-1.6>>>

      <strong|Answer:> Taking <math|<dfrac|d|d<hspace|0pt>x>> on both sides
      of the equation (suppose <math|y> is a function of <math|x>):

      <\equation*>
        <dfrac|d|d<hspace|0pt>x>*<around*|(|x<rsup|2>+y<rsup|2>|)>=<dfrac|d|d<hspace|0pt>x>*<around*|(|25|)>
      </equation*>

      <\equation*>
        <dfrac|d|d<hspace|0pt>x>*x<rsup|2>+<dfrac|d|d<hspace|0pt>x>*y<rsup|2>=0
      </equation*>

      <\equation*>
        2*x+2*y*<with|color|blue|<dfrac|d<hspace|0pt>y|d<hspace|0pt>x>>=0
        <text|by the chain rule><space|1em>\<Rightarrow\><space|1em><dfrac|d<hspace|0pt>y|d<hspace|0pt>x>=-<dfrac|x|y>=-<dfrac|3|4>.
      </equation*>

      For this example, we can solve <math|y> for <math|x> and double check
      the answer.

      <\equation*>
        y=\<pm\><sqrt|25-x<rsup|2>>=<sqrt|25-x<rsup|2>>
        near**<around*|(|3,4|)>
      </equation*>

      <\equation*>
        <dfrac|d<hspace|0pt>y|d<hspace|0pt>x>=<dfrac|-2*x|2<sqrt|25-x<rsup|2>>>=-<dfrac|x|<sqrt|25-x<rsup|2>>>=-<dfrac|3|4>.
      </equation*>

      \;

      <\example>
        Lemniscate curve:

        <\wide-tabular>
          <tformat|<cwith|1|1|1|1|cell-halign|c>|<table|<row|<\cell>
            <\script-output|asymptote|default>
              % -width 0.5par

              import contour;

              size(10cm, 0);

              draw((-3,0)--(3,0), arrow=Arrow, gray+linewidth(0.5pt));

              label("$x$", (3,0), align=E);

              draw((0,-1.5)--(0,1.5), arrow=Arrow, gray+linewidth(0.5pt));

              label("$y$", (0,1.5), align=N);

              real f(real x, real y) {return (x^2+y^2)^2-4*(x^2-y^2);}

              guide[][] thegraphs = contour(f, a=(-2,-2), b=(2,2), new real[]
              {0});

              draw(thegraphs[0], linewidth(1pt));

              label("$(x^2+y^2)^2=4(x^2-y^2)$", (0,1), align=E);
            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            \;
          </cell>>>>
        </wide-tabular>

        <\equation*>
          <dfrac|d|d<hspace|0pt>x>*<around*|(|x<rsup|2>+y<rsup|2>|)><rsup|2>=<dfrac|d|d<hspace|0pt>x>
          <around*|[|4*<around*|(|x<rsup|2>-y<rsup|2>|)>|]>
        </equation*>

        <\equation*>
          2*<around*|(|x<rsup|2>+y<rsup|2>|)>*<around*|(|2*x+2*y*y<rprime|'>|)>=4*<around*|(|2x-2*y*y<rprime|'>|)>
        </equation*>

        <\equation*>
          \<Rightarrow\><space|1em>y<rprime|'>=<dfrac|2x-x*<around*|(|x<rsup|2>+y<rsup|2>|)>|y*<around*|(|x<rsup|2>+y<rsup|2>|)>+2*y>
        </equation*>
      </example>

      <section|Derivatives of Logarithmic Functions>

      <\eqnarray*>
        <tformat|<table|<row|<cell|y>|<cell|=>|<cell|log<rsub|b>
        x>>|<row|<cell|x>|<cell|=>|<cell|b<rsup|y>>>|<row|<cell|<dfrac|d|d<hspace|0pt>x>
        x>|<cell|=>|<cell|<dfrac|d|d<hspace|0pt>x>
        b<rsup|y>>>|<row|<cell|1>|<cell|=>|<cell|<around*|(|ln
        b|)>b<rsup|y>*y<rprime|'>>>|<row|<cell|\<Rightarrow\><space|1em>y<rprime|'>>|<cell|=>|<cell|<dfrac|1|<around*|(|ln
        b|)>b<rsup|y>>>>|<row|<cell|>|<cell|=>|<cell|<dfrac|1|<around*|(|ln
        b|)>*x>>>>>
      </eqnarray*>

      <\theorem>
        <dueto|Derivative of logarithmic functions>

        <\equation*>
          <around*|(|log<rsub|b> x|)><rprime|'>=<dfrac|1|<around*|(|ln
          b|)>*x>,<space|1em><around*|(|ln x|)><rprime|'>=<dfrac|1|x>.
        </equation*>
      </theorem>

      \;

      <subsection|Logarithmic differentiation>

      Using the properties

      <\equation*>
        ln <around*|(|x*y|)>=ln x+ln y,<space|1em>ln x<rsup|y>=y*ln x
      </equation*>

      and implicit differentiation, one can simplify the differentiation of
      some functions.

      <\example>
        Find <math|<dfrac|d<hspace|0pt>y|d<hspace|0pt>x>> if
        <math|y=<dfrac|<around*|(|x<rsup|2>+1|)><rsup|5>*<around*|(|x-1|)><rsup|3>|<sqrt|x<rsup|2>+sin
        x+2>>>.
      </example>

      <strong|Answer:> First, take <math|ln> on both sides.

      <\eqnarray*>
        <tformat|<table|<row|<cell|ln y>|<cell|=>|<cell|ln
        <dfrac|<around*|(|x<rsup|2>+1|)><rsup|5>*<around*|(|x-1|)><rsup|3>|<sqrt|x<rsup|2>+sin
        x+2>>>>|<row|<cell|>|<cell|=>|<cell|5*ln<around*|(|x<rsup|2>+1|)>+3*ln
        <around*|(|x-1|)>-<dfrac|1|2>*ln <around*|(|x<rsup|2>+sin x+2|)>>>>>
      </eqnarray*>

      Then take <math|<dfrac|d|d<hspace|0pt>x>> on both sides,

      \;

      <\eqnarray*>
        <tformat|<table|<row|<cell|<dfrac|1|y>*y<rprime|'>>|<cell|=>|<cell|<dfrac|5|x<rsup|2>+1>\<cdot\>2**x+<dfrac|3|x-1>-<dfrac|1|2>\<cdot\><dfrac|2*x+cos
        x|x<rsup|2>+sin x+2>>>|<row|<cell|>|<cell|=>|<cell|<dfrac|10*x|x<rsup|2>+1>+<dfrac|3|x-1>-<dfrac|2*x+cos
        x|2*<around*|(|x<rsup|2>+sin x+2|)>>>>|<row|<cell|\<Rightarrow\>>|<cell|>|<cell|y<rprime|'>=y*<around*|(|<dfrac|10*x|x<rsup|2>+1>+<dfrac|3|x-1>-<dfrac|2*x+cos
        x|2*<around*|(|x<rsup|2>+sin x+2|)>>|)>>>|<row|<cell|>|<cell|>|<cell|=<dfrac|<around*|(|x<rsup|2>+1|)><rsup|5>*<around*|(|x-1|)><rsup|3>|<sqrt|x<rsup|2>+sin
        x+2>><around*|(|<dfrac|10*x|x<rsup|2>+1>+<dfrac|3|x-1>-<dfrac|2*x+cos
        x|2*<around*|(|x<rsup|2>+sin x+2|)>>|)>>>>>
      </eqnarray*>

      <\remark>
        Let <math|z=ln y>, then

        <\equation*>
          <dfrac|d*z|d<hspace|0pt>x>=<dfrac|d*z|d<hspace|0pt>y>*<dfrac|d<hspace|0pt>y|d<hspace|0pt>x>=<dfrac|1|y>*y<rprime|'>.
        </equation*>
      </remark>

      <\example>
        <math|y=f<around*|(|x|)>=x<rsup|x>>.
      </example>

      <strong|Answer:>\ 

      <\eqnarray*>
        <tformat|<table|<row|<cell|ln y>|<cell|=>|<cell|x*ln
        x>>|<row|<cell|<dfrac|d|d<hspace|0pt>x> <around*|(|ln
        y|)>>|<cell|=>|<cell|<dfrac|d|d<hspace|0pt>x> <around*|(|x*ln
        x|)>>>|<row|<cell|<dfrac|y<rprime|'>|y>>|<cell|=>|<cell|ln
        x+x*<dfrac|1|x>=1+ln x>>|<row|<cell|y<rprime|'>>|<cell|=>|<cell|x<rsup|x>*<around*|(|1+ln
        x|)>>>>>
      </eqnarray*>

      <section|Rates of Change in the Economics and Social Sciences>

      Read the slides.

      \;

      <\section>
        skip
      </section>

      <section|skip>

      <section|Linear Approximations and Differentials>

      <subsection|Linear approximations>

      \;

      Suppose <math|f> is differentiable at <math|a>, then

      <\equation*>
        f<rprime|'><around*|(|a|)>=lim<rsub|x\<rightarrow\>a>
        <dfrac|f<around*|(|x|)>-f<around*|(|a|)>|x-a>\<approx\><dfrac|f<around*|(|x|)>-f<around*|(|a|)>|x-a><space|1em><text|when
        >x <text|is close to >a
      </equation*>

      <\equation*>
        \<Rightarrow\><space|1em>f<around*|(|x|)>-f<around*|(|a|)>\<approx\>f<rprime|'><around*|(|a|)>*<around*|(|x-a|)>
      </equation*>

      <\equation*>
        <marked|f<around*|(|x|)>\<approx\>f<around*|(|a|)>+f<rprime|'><around*|(|a|)>*<around*|(|x-a|)>>.
      </equation*>

      is called the <strong|linear approximation> of <math|f<around*|(|x|)>>.
      And the linear function

      <\equation*>
        L<around*|(|x|)>=f<around*|(|a|)>+f<rprime|'><around*|(|a|)>*<around*|(|x-a|)>
      </equation*>

      is called the <strong|linearization> of <math|f<around*|(|x|)>>.

      \;

      <\wide-block>
        <tformat|<cwith|1|-1|1|-1|cell-halign|l>|<cwith|1|-1|1|-1|cell-valign|t>|<cwith|1|-1|1|-1|cell-rsep|1spc>|<cwith|1|-1|1|-1|cell-bsep|1spc>|<cwith|1|-1|1|-1|cell-tsep|1spc>|<cwith|1|-1|1|-1|cell-lsep|1spc>|<table|<row|<\cell>
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        </cell>|<\cell>
          <\example>
            Estimate the value of <math|<sqrt|4.1>> and <math|<sqrt|3.99>>
            using a linear approximation.
          </example>

          <strong|Answer:> Let <math|f<around*|(|x|)>=<sqrt|x>>. Then
          <math|f<rprime|'>=<dfrac|1|2*<sqrt|x>>>

          <\equation*>
            f<around*|(|4.1|)>\<approx\>f<around*|(|4|)>+f<rprime|'><around*|(|4|)>*0.1=2+<dfrac|0.1|4>=2.025.
          </equation*>

          <\equation*>
            f<around*|(|3.99|)>\<approx\>f<around*|(|4|)>+f<rprime|'><around*|(|4|)>*<around*|(|-0.01|)>=2-<dfrac|0.01|4>=1.9975
          </equation*>

          \;
        </cell>>>>
      </wide-block>

      \;

      <\example>
        Approximate the value of <math|sin<around*|(|0.2|)>>.
      </example>

      <strong|Answer:> Let <math|f<around*|(|x|)>=sin x>, then
      <math|f<rprime|'><around*|(|x|)>=cos x>,\ 

      <\equation*>
        f<around*|(|0.2|)>\<approx\>f<around*|(|0|)>+f<rprime|'><around*|(|0|)>*0.2=0+1\<times\>0.2=0.2
      </equation*>

      <\session|python|default>
        <\output>
          Python 3.9.6 [/Library/Developer/CommandLineTools/usr/bin/python3]\ 

          Python plugin for TeXmacs.

          Please see the documentation in Help -\<gtr\> Plugins -\<gtr\>
          Python
        </output>

        <\unfolded-io>
          \<gtr\>\<gtr\>\<gtr\>\ 
        <|unfolded-io>
          4.1**0.5
        <|unfolded-io>
          2.0248456731316584
        </unfolded-io>

        <\unfolded-io>
          \<gtr\>\<gtr\>\<gtr\>\ 
        <|unfolded-io>
          3.99**0.5
        <|unfolded-io>
          1.997498435543818
        </unfolded-io>

        <\unfolded-io>
          \<gtr\>\<gtr\>\<gtr\>\ 
        <|unfolded-io>
          math.sin(0.2)
        <|unfolded-io>
          0.19866933079506122
        </unfolded-io>

        <\input>
          \<gtr\>\<gtr\>\<gtr\>\ 
        <|input>
          \;
        </input>
      </session>

      \;

      <\example>
        Find an approximate value of <math|<sqrt|29|3>>.
      </example>

      <strong|Answer:> Let <math|f<around*|(|x|)>=<sqrt|x|3>>. Then
      <math|f<rprime|'><around*|(|x|)>=<dfrac|1|3>*x<rsup|-<dfrac|2|3>>>.\ 

      <\equation*>
        f<around*|(|29|)>\<approx\>f*<around*|(|27|)>+f<rprime|'><around*|(|27|)>*<around*|(|29-27|)>=3+<dfrac|2|27>=<dfrac|83|27>.
      </equation*>

      <\session|python|default>
        <\unfolded-io>
          \<gtr\>\<gtr\>\<gtr\>\ 
        <|unfolded-io>
          29**(1/3)
        <|unfolded-io>
          3.072316825685847
        </unfolded-io>

        <\unfolded-io>
          \<gtr\>\<gtr\>\<gtr\>\ 
        <|unfolded-io>
          83/27
        <|unfolded-io>
          3.074074074074074
        </unfolded-io>
      </session>

      <subsection|Differentials>

      Suppose <math|y=f<around*|(|x|)>> is differentiable at <math|x>, then

      <\equation*>
        <dfrac|d<hspace|0pt>y|d<hspace|0pt>x>=f<rprime|'><around*|(|x|)><space|1em>\<Rightarrow\><space|1em>d<hspace|0pt>y=f<rprime|'><around*|(|x|)>*d<hspace|0pt>x,
      </equation*>

      where we call <math|d<hspace|0pt>x,d<hspace|0pt>y>
      <strong|differentials>.

      <\example>
        The radius of a sphere was measured and found to be
        <math|<rigid|21<space|0.6spc>cm>> with a possible error in
        measurement of at most <rigid|<math|0.05<space|0.6spc>cm>>. Estimate
        the maximum error in using this value of the radius to compute the
        volume of the sphere?
      </example>

      <strong|Answer:> Let <math|r,V> denote the radius and volume. Then

      <\equation*>
        V=<dfrac|4|3>*\<pi\>*r<rsup|3>.
      </equation*>

      Then <math|V<rprime|'>=4*\<pi\>*r<rsup|2>>

      <\equation*>
        d V=V<rprime|'><around*|(|r|)>*d*r=4*\<pi\>*r<rsup|2>*d*r
      </equation*>

      Then

      <\equation*>
        <around*|\||d*V|\|>=4*\<pi\>*r<rsup|2>*<around*|\||d*r|\|>\<leqslant\>4*\<pi\>*21<rsup|2>*0.05\<approx\>277*cm<rsup|3>.
      </equation*>

      The relative change of the volume is

      <\equation*>
        <around*|\||<dfrac|d*V|V>|\|>=<dfrac|3|r><around*|\||d*r|\|>\<leqslant\><dfrac|0.05|7>\<approx\>0.007=7%.
      </equation*>
    </slide>
  </slideshow>
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    <associate|auto-16|<tuple|9|9>>
    <associate|auto-17|<tuple|10|9>>
    <associate|auto-18|<tuple|10.1|9>>
    <associate|auto-19|<tuple|10.2|10>>
    <associate|auto-2|<tuple|1.1|1>>
    <associate|auto-3|<tuple|1.2|2>>
    <associate|auto-4|<tuple|1.3|2>>
    <associate|auto-5|<tuple|2|3>>
    <associate|auto-6|<tuple|2.1|3>>
    <associate|auto-7|<tuple|2.2|4>>
    <associate|auto-8|<tuple|3|4>>
    <associate|auto-9|<tuple|3|4>>
  </collection>
</references>

<\auxiliary>
  <\collection>
    <\associate|toc>
      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|1<space|2spc>Derivatives
      of Polynomials and Exponential Functions>
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-1><vspace|0.5fn>

      <with|par-left|<quote|1tab>|1.1<space|2spc>Power functions
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-2>>

      <with|par-left|<quote|1tab>|1.2<space|2spc>Linear combination
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-3>>

      <with|par-left|<quote|1tab>|1.3<space|2spc>Exponential functions
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-4>>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|2<space|2spc>The
      Product and Quotient Rules> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-5><vspace|0.5fn>

      <with|par-left|<quote|1tab>|2.1<space|2spc>The product rule
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-6>>

      <with|par-left|<quote|1tab>|2.2<space|2spc>The quotient rule
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-7>>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|3<space|2spc>Derivatives
      of Trigonometric Functions> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-8><vspace|0.5fn>

      <with|par-left|<quote|1tab>|Derivative of
      <with|color|<quote|black>|font-family|<quote|rm>|<with|math-display|<quote|true>|<with|mode|<quote|math>|<with|font-base-size|<quote|12>|sin
      x>>>> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-9>>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|4<space|2spc>The
      Chain Rule> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-10><vspace|0.5fn>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|5<space|2spc>Implicit
      differentiation> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-11><vspace|0.5fn>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|6<space|2spc>Derivatives
      of Logarithmic Functions> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-12><vspace|0.5fn>

      <with|par-left|<quote|1tab>|6.1<space|2spc>Logarithmic differentiation
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-13>>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|7<space|2spc>Rates
      of Change in the Economics and Social Sciences>
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-14><vspace|0.5fn>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|8<space|2spc>skip>
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-15><vspace|0.5fn>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|9<space|2spc>skip>
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-16><vspace|0.5fn>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|10<space|2spc>Linear
      Approximations and Differentials> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-17><vspace|0.5fn>

      <with|par-left|<quote|1tab>|10.1<space|2spc>Linear approximations
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-18>>

      <with|par-left|<quote|1tab>|10.2<space|2spc>Differentials
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-19>>
    </associate>
  </collection>
</auxiliary>